Context
In my lecture notes on tensor categories it says:
"For a given category $C$ and a given tensor product $\otimes$, inequivalent associators can exist."
Questions
What notion of equivalence is (usually) meant here?
Is it simply the equality of two natural transformations?
That is, does the proposition read 'There exist monoidal categories $(C, \otimes, I, a_1, l_1, r_1)$ and $(C, \otimes, I, a_2, l_2, r_2)$ with associators $a_2 \neq a_1$'?What are examples of such categories with inequivalent associators?
Take the category $Vect^\mathbb{Z}_\mathbb{K}$ of $\mathbb{Z}$ graded $\mathbb{K}$-vector spaces with the graded tensor product: $$ (V\otimes W)_n=\bigoplus_{i+j = n}(V_i\otimes W_{j}). $$
There's the usual associator: $$ (a\otimes b)\otimes c \in(U\otimes V)\otimes W\mapsto a\otimes (b\otimes c)\in U\otimes (V\otimes W). $$
Another associator takes the grading into account: $$ (a\otimes b)\otimes c \in(U\otimes V)\otimes W \mapsto (-1)^{i+k} a\otimes (b\otimes c)\in U\otimes (V\otimes W), $$ where $i$ and $k$ are the gradings of $a$ and $c$, respectively. The index $j$ of $b$ was omitted for the pentagon axiom to work.
The monoidal categories defined by these associators aren't monoidally equivalent. In fact, a function $a:\mathbb{Z}^3\to\mathbb{K}^*$ defines an associator for $Vect^\mathbb{Z}_\mathbb{K}$ iff $$ a(r,s,t)a(r,st,v)a(s,t,v)a(r,s,tv)^{-1}a(rs,t,v)^{-1} = 1 $$ for all $r,s,t,v\in\mathbb{Z}$. This is the same as saying that $a$ is a nontrivial 3-cocycle of $\mathbb{Z}$ with coefficients in $\mathbb{K}^*$.
For more details you can see Example 1.7 of these lectures notes.
edit: An example very similar to this one is worked out in detail at Kerodon: the monoidal structure of a 3-cocycle is defined in Example 2.1.3.3, and in Example 2.1.6.8 it's proven that 3-chains $a,a'$ define equivalent monoidal structures if and only if they are cohomologous. This is also revisited at Example 2.1.15.